(-1.0, 4.0)
(-1.0, 4.0)
Ingeniería Biomédica
2025-03-10
Convolution is a fundamental operation in signal processing.
Given two signals ( x(t) ) and ( h(t) ), their convolution is defined as:
\[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau \]
In discrete-time, convolution is:
\[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]
Convolution in time domain corresponds to multiplication in frequency domain:
\[ X(f) H(f) = Y(f) \]
This property is crucial in filter design and system analysis.
(-1.0, 4.0)
(-1.0, 4.0)
Convolution requiere the representation of the signal in a sum of impulse functions.
Fourier series represents periodic signals as a sum of sinusoids:
\[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t} \]
where ( C_n ) are the Fourier coefficients.
Decomposing a signal into sinusoidal components allows frequency analysis.
The Fourier coefficients ( C_n ) are computed as:
\[ C_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt \]
Determines how much of each frequency is present in the signal.
(array([-5., 0., 5., 10., 15., 20., 25., 30., 35.]), [Text(-5.0, 0, '−5'), Text(0.0, 0, '0'), Text(5.0, 0, '5'), Text(10.0, 0, '10'), Text(15.0, 0, '15'), Text(20.0, 0, '20'), Text(25.0, 0, '25'), Text(30.0, 0, '30'), Text(35.0, 0, '35')])
(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])
(array([-2., 0., 2., 4., 6., 8., 10., 12., 14., 16.]), [Text(-2.0, 0, '−2'), Text(0.0, 0, '0'), Text(2.0, 0, '2'), Text(4.0, 0, '4'), Text(6.0, 0, '6'), Text(8.0, 0, '8'), Text(10.0, 0, '10'), Text(12.0, 0, '12'), Text(14.0, 0, '14'), Text(16.0, 0, '16')])
(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])
Sampling theorem: A signal must be sampled at a frequency at least twice its highest frequency component:
\[ f_s \geq 2 f_{max} \]
Aliasing occurs if sampling frequency is too low.